Table Of ContentCoalescing binaries as possible standard candles
Salvatore Capozziello, Mariafelicia De Laurentis, Ivan De Martino, Michelangelo Formisano
Dipartimento di Scienze Fisiche, Universita` di Napoli “Federico II”, INFN Sez. di Napoli,
Compl. Univ. di Monte S. Angelo, Edificio G, Via Cinthia, I-80126, Napoli, Italy
(Dated: January 9, 2010)
Gravitational waves detected from well-localized inspiraling binaries would allow to determine,
directly and independently, both binary luminosity and redshift. In this case, such systems could
behave as ”standard candles” providing an excellent probe of cosmic distances up to z < 0.1 and
thuscomplementing other indicators of cosmological distance ladder.
0
Keywords: gravitationalwaves, standard candles, cos- one is called coalescing time, interesting for detectors as
1
0 mological distances theAmericanLIGO(LaserInterferometerGravitational-
2 WaveObservatory)[12]andFrench/ItalianVIRGO [13].
Coalescence is a rare event and therefore to see several
n
events per year, LIGO and/or VIRGO must look far be-
a
I. INTRODUCTION
J yond our Galaxy. For example, the expected rate for
9 a NS-NS system (determined from the observed popu-
A new type of standard candles, or, more appropri- lation of NS-NS binaries) is found to be 80+210 Myr−1
−70
] ately, standard sirens,could be achievedby studying co- per galaxy [14]. From this figure, one finds that the ex-
O
alescingbinarysystems[1,2]. Thesesystems areusually pected rate for the today available sensitivities of today
C considered strong emitter of gravitational waves (GW), LIGO and VIRGO is of the order 35+90 × 10−3 yr−1,
−30
. ripples of space-time due to the presence of accelerated while for the advanced version of such interferometers,
h
p masses in analogy with the electromagnetic waves, due the rate is more interesting being 190+−417500 yr−1, that is
- to accelerated charged. The coalescence of astrophysical the probability ranges from one event per week to two
o systems containing relativistic objects as neutron stars events per day. A remarkable fact about binary coales-
r
t (NS), white dwarves (WD) and black holes (BH) consti- cence is that it can provide an absolute measurement of
s
tuteverystandardGWsourceswhichcouldbeextremely thesourcedistance: thisisanextremelyimportantevent
a
[ usefulforcosmologicaldistanceladderifphysicalfeatures in Astronomy. In fact, for these systems, the distance
ofGWemissionarewelldetermined. Thesebinariessys- is given by the measure of the GW polarization emit-
2
tems, as the famous PSR 1913+16 [3–5], have compo- ted during the coalescence. One of the problems which
v
nents that are gradually inspiralling one over the other affects the utilization of coalescing binaries as standard
1
6 as the result of energy and angular momentum loss due candlesisthemeasureoftheredshiftofthesourceaswell
9 to (also) gravitational radiation. As a consequence the as the measure of the GW polarization (at present GWs
0 GW frequency is increasing and, if observed, could con- have not been still experimentally observed). A solution
8. stitutea”signature”forthewholesystemdynamics. The for the redshift determination could be the detection of
0 coalescence of a compact binary system is usually clas- an electromagnetic counterpart of the coalescing system
9 sified in three stages, which are not very well delimited (e.g. the detection of an associated gamma ray burst)
0 one from another, namely the inspiral phase, the merger orthe redshiftmeasurementofthehostgalaxyorgalaxy
v: phase and the ring-down phase. The merger phase is clusterattheirbarycenter[15]. Recentevidencesupports
i the process that proceeds until the collision of the bod- the hypothesis that many short-hard gamma-ray bursts
X
ies and the formation of a unique object. Its duration could be associated with coalescing binary systems in-
r depends on the characteristics of the originating stars deed [16–19]. In this paper, we want to show that such
a
and emission is characterized by a frequency damping systems could be used as reliable standard candles.
in the time. In merger phase, stars are not modelled
as rigid sphere due to the presence of a convulsive ex- In Sect. II, we briefly sketch the GW emission from
change of matter [6]. GW emission from merger phase coalescing binary systems in circular orbit at cosmologi-
can only be evaluated using the full Einstein equations. cal distance. In Sect. III, we simulate various coalescing
Because of the extreme strong field nature of this phase, binary systems (WD-WD, NS-NS, BH-BH) at redshift
neither a straightforward application of post-Newtonian z < 0.1 because of the observational limits of ground-
theory nor any perturbation theory is very useful. Re- based-interferometersas LIGO and VIRGO. The goalof
cent numerical work [7–9] has given some insight into thesimulationisthemeasureoftheHubbleconstantand
the mergerproblem, but there are no reliable models for consequently the use of these systems as standard can-
the waveform of the merger phase up to now. Gravita- dles [20]. This new type of standard candles will be able
tional radiation from the ring-down phase is well known toincreasetheconfidencelevelontheother”traditional”
anditcanbedescribedbyquasi-normalmodes[10]. The standard candles in Astronomy and, moreover, it could
relevance of ring-down phase is described in [11]. Tem- constituteaneffectivetooltomeasuredistancesatlarger
poralintervalbetweenthe inspiralphase andthe merger redshifts. Conclusions are drawn in Sect. IV.
2
II. GRAVITATIONAL RADIATION FROM A
z z
COALESCING BINARY SYSTEM
y y
For a detailed exposition of GW theory see, e.g. [21–
23]. Letusconsiderhereanisolated,farawayandslowly
moving source. In this approximation, we can write the
GW solution as:
x x
2Gd2I
h¯ (t,x)= ij(t ), (1) Figure 1: Lines of force associated to the + (left panel) and
µν r dr2 R × (right panel) polarizations.
where I is the quadrupole momentum tensor of the en-
ij
ergy density of source, conventionally defined as:
Φ is given by the expression:
I (t)= yiyjT00(t,y)d3y, (2) t
ij
Z Φ(t)=2π dt′f(t′). (6)
Z
a tensor defined at any constant time surface, and tR is t0
the retardedtime [21]. The distance between source and
Note that for a fixed distance r and a given frequency
observer is denoted as r.
f, the GW amplitudes are fully determined by µM2/3 =
The gravitational wave produced by an isolated non-
M 5/3, where the combination:
relativistic object is therefore proportionalto the second C
derivative of the quadrupole momentum of the energy
dobensesritvyeraitnttehresepctosintthewshoeurercet.heInpacsotntlirgahstt,ctohneeleoafdtinhge M = (M1M2)3/5 , (7)
contributiontoelectromagneticradiationcomesfromthe C (M1+M2)1/5
changing dipole momentum of the charge density.
The aboveresultcanbe specifiedfortwopoint masses is called chirp mass of the binary, here M = M1 +M2
M1 and M2 in a circular orbit. In the quadrupole ap- is the total mass of the system and µ= M1M2 is the
proximation,the two polarizationamplitudes of GWs at M1+M2
reducedmassofthe system. Introducingthe coalescence
a distance r from the source are given evaluating Eq.(1)
time τ =t −t and integrating Eq. (5), we get:
to lowest order in v/c, that is: coal
4 GM 5/3 πf(t ) 2/3 1+cos2i f ≃130 1.21M⊙ 5/8 1sec 3/8Hz. (8)
h+(t)= r (cid:18) c2C(cid:19) (cid:18) c R (cid:19) (cid:18) 2 (cid:19)× (cid:18) MC (cid:19) (cid:18) τ (cid:19)
Eq. (8) predicts coalescence times of τ ∼
17min, 2sec, 1msec for f ∼10, 100,1000Hz.
cos[Φ(t )] ,
R After averaging over the orbital period and the ori-
(3)
entations of the binary orbital plane, one arrives at the
average (characteristic) GW amplitude:
4 GM 5/3 πf(t ) 2/3
C R
h×(t)= r (cid:18) c2 (cid:19) (cid:18) c (cid:19) cosisin[Φ(tR)], h(f,M ,r)= h2 + h2 1/2 =
C + ×
(4) (cid:0)(cid:10) (cid:11) (cid:10) (cid:11)(cid:1)
(9)
where i is the binary inclination angle such that i = 32 1/2 G5/3M5/3
90◦ corresponds to a system visible edge-on. These are = C (πf)2/3 .
(cid:18) 5 (cid:19) c4 r
traditionallylabeled”plus”and”cross”fromthe linesof
force associatedwith their tidal stretch and squeeze (see
For a binary at the cosmological distance, i.e. at red-
Fig. 1). Here f is the frequency of the emitted GWs
shiftz whereGWspropagateinaFriedmann-Robertson-
(twice the orbital frequency). The rate of the frequency
Walker Universe, these equations are modified in a very
change is [22]:
straightforwardway:
• The frequency that appears in the above formulae
5/3
96 GM is the frequency measured by the observer, f ,
f˙= π8/3 C f11/3, (5) obs
5 (cid:18) c3 (cid:19) which is red-shifted with respect to the source fre-
quency f , i.e. f = f /(1+z), and similarly t
s obs s
where t is the so called ”retarded time” and the phase and t are measured with the observer clocks.
R R
3
• The chirp mass M has to be replaced by M = where H(z) is the value of the Hubble parameterat red-
C C
M (1+z). shift z. Knowing d (z), we can therefore obtain H(z).
C L
This shows that the luminosity distance function d (z)
L
• The distance r to the source has to be replaced by is an extremely important quantity, which encodes the
the luminosity distance dL(z). whole expansion history of the Universe. Coalescing bi-
naries could be standard candles (or precisely standard
Inserting the following quantity:
sirens)in the following sense. Suppose that we can mea-
suretheamplitudesofbothpolarizationsh+,h×,aswell
4 GMC(z) 5/3 πf(t) 2/3 as f˙obs. From the ratio of h+ and h×, we can obtain
h (t)= , (10) the value of the inclination of the orbit, besides, evalu-
c dL(z)(cid:18) c2 (cid:19) (cid:18) c (cid:19) ating f˙ at a given frequency, we can obtain M . If
obs C
we are capable of measuring the redshift z of the source,
we can rewrite the expressions for the polarization ”+”
and ”×” as: we have found a gravitational standard candle since we
can obtain the luminosity distance from Eq. (13) and
then evaluate the Hubble constant H0. The difference
1+cos2i between gravitational standard candles and the ”tradi-
h+(t)=hc(tR) cos[Φ(tR)] , (11) tional” standard candles is that the luminosity distance
2
is directly linked to the GW polarization and there is
and no theoretical uncertainty on its determination a part
the redshift evaluation. Various possibilities have been
h×(t)=hc(tR)cosisin[Φ(tR)] . (12) proposed. Among these there is the possibility to see
an optical counterpart. In fact, it can be shown that
Explicating the dependence on the chirp mass redshift, observations of the GWs emitted by inspiralling binary
wecanobtaintheluminositydistancedL usingtheequa- compactsystemscanbeapowerfulprobeatcosmological
tion (10) linked directly to the GW polarization: scales. In particular, short GRBs appear related to such
systems and quite promising as potential GW standard
sirens[34]). Ontheotherhand,theredshiftofthebinary
5/3
d (z)= 4 GMC(1+z) [πf(t)]2/3 = system can be associated to the barycenter of the host
L hc(t)(cid:20) c2 (cid:21) galaxy or the galaxy cluster as we are going to do here.
(13)
4(1+z)5/3 GM 5/3
= C [πf(t)]2/3 .
h (t) (cid:20) c2 (cid:21) III. NUMERICAL SIMULATION
c
Let us recall that the luminosity distance d of a source
L We have simulated various coalescing binary systems
is defined by
at redshifts z < 0.1 In this analysis, we do not consider
L systematic errors and errors on redshifts to obviate the
F = , (14) absence of a complete catalogue of such systems. The
4πd2
L choice of low redshifts is due to the observational limits
of ground-based interferometers like VIRGO or LIGO.
where F is the flux (energy per unit time per unit area)
Some improvements are achieved, if we take into ac-
measuredbytheobserver,andListheabsoluteluminos-
count the future generation of these interferometers as
ityofthesource,i.e. thepowerthatitradiatesinitsrest
Advanced VIRGO [35]and Advanced LIGO [36]. Ad-
frame. For small redshifts, d is related to the present
L vanced VIRGO is a major upgrade, with the goal of
valueoftheHubbleparameterH0andtothedeceleration increasing the sensitivity by about one order of mag-
parameter q0 by nitude with respect to VIRGO in the whole detection
band. Such a detector, with Advanced LIGO, is ex-
H0dL =z+ 1(1−q0)z2+... . (15) pected to see many events every year (from 10s to 100s
c 2
events/year). For example, a NS-NS coalescence will be
The first term of this expansion gives the Hubble law detectable as far as 300 Mpc. In the simulation pre-
z ≃(H0/c)dL, which states that redshift is proportional sented here, sources are slightly out of LIGO-VIRGO
to the distance. The term O(z2) is the correction to the band but observable, in principle, with future interfer-
linear law for moderate redshifts. For large redshifts, ometers. Here, we have used the redshifts taken by
the Taylorseries is no longerappropriate,and the whole NASA/IPACEXTRAGALACTICDATABASE[37],and
expansionhistoryoftheUniverseisencodedinafunction wehavefixedtheredshiftusingzatthebarycenterofthe
dL(z). For a spatially flat Universe, one finds hostgalaxy/cluster),andthebinarychirpmass MC,typ-
ically measured, from the Newtonian part of the signal
z dz′ at upward frequency sweep, to ∼0.04% for a NS/NS bi-
d (z)=c(1+z) , (16)
L Z0 H(z′) naryand∼0.3%forasystemcontainingatleastoneBH
4
[28, 38]. The distance to the binary d (”luminosity dis-
L
tance” at cosmological distances) can be inferred, from
the observed waveforms, to a precision ∼ 3/ρ . 30%,
where ρ = S/N is the amplitude signal-to-noise ratio in
the total LIGO network (which must exceed about 8 in
orderthatthe falsealarmratebe lessthanthe threshold
for detection). In this way, we have fixed the character-
istic amplitude of GWs, and frequencies are tuned in a
range compatible with such a fixed amplitude, then the
errorondistanceluminosityiscalculatedbythe erroron
the chirp mass with standard error propagation.
The systems considered are NS-NS, BH-BH and WD-
WD. For eachof them, a particular frequency range and
a characteristic amplitude (beside the chirp mass) are
fixed. We start with the analysis of NS-NS systems
(MC = 1.22M⊙) with characteristic amplitude fixed to
the value 10−22. In Table I, we report the redshift, the
valueofh andthefrequencyrangeofsystemsanalyzed.
C
Figure 2: Luminosity distance vs redshift for simulated NS-
Object z hc Freq. (Hz)
NS systems.
NGC 5128 0.0011 10−22 0÷10
NGC 1023 Group 0.0015 10−22 0÷10
NGC 2997 0.0018 10−22 5÷15
NGC 5457 0.0019 10−22 10÷20
NGC 5033 0.0037 10−22 25÷35
Virgo Cluster 0.0042 10−22 30÷40
Fornax Cluster 0.0044 10−22 35÷45
NGC 7582 0.0050 10−22 45÷55
Ursa Major Groups 0.0057 10−22 50÷60
EridanusCluster 0.0066 10−22 55÷65
TableI:Redshifts,characteristicamplitudes,frequencyrange
for NS-NSsystems.
In Fig. 2, the derivedHubble relationis reported, and
in Fig. 3 residuals are reported.
The Hubble constant value is 72 ± 1 km/sMpc in
agreement with the recent WMAP estimation (Wilkin-
sonMicrowaveAnisotropy Probe [25]). The same proce-
dure is adopted for WD-WD systems (MC = 0.69M⊙, Figure3: Residualsoftheplotluminositydistancevsredshift
hC = 10−23) and BH-BH systems (MC = 8.67M⊙, for simulated NS-NSsystems.
h =10−21). InTablesIIandIII,wereporttheredshift,
C
the value of h and the frequency range for BH-BH and
C
WD-WD systems respectively. These simulations are re- barycenter of the host astrophysical system as galaxy,
portedinFig. 4andinFig. 5: the correspondingresidu- group of galaxies or cluster of galaxies. In such a way,
als have been checked and their goodness is the same of the standard methods adopted to evaluate the cosmic
previous case so we omit them. distances (e.g. Tully-Fisher or Faber-Jackson relations)
For these simulations, the Hubble constant value is can be considered as ”priors” to fit the Hubble rela-
69 ± 2 km/sMpc and 70 ± 1 km/sMpc for BH-BH tion. We have simulated various situations assuming
and WD-WD systems respectively, also in these cases NS-NS, BH-BH, and WD-WD binary systems. Clearly,
in agreement with WMAP estimation. the leading parameter is the chirp mass M , or its red-
C
shiftedcounter-partM ,whichisdirectlyrelatedtothe
C
GW amplitude. The adopted redshifts are in a well-
IV. CONCLUSIONS testedrangeofscalesandtheHubbleconstantvalueisin
good agreement with WMAP estimation. The Hubble-
Inthispaper,wehaveconsideredsimulatedbinarysys- luminosity-distance diagrams of the above simulations
tems whose redshifts can be estimated considering the showthe possibility to use the coalescingbinary systems
5
Object z hc Freq. (Hz) Object z hc Freq. (Hz)
Pavo-IndusSup.Cluster 0.015 10−21 65÷70 Eridanus Cluster 0.0066 10−23 5÷10
Abell 569 Cluster 0.019 10−21 75÷80 HydraCluster 0.010 10−23 15÷20
Coma Cluster 0.023 10−21 100÷105 Payo-IndusSup.Cluster 0.015 10−23 35÷40
Abell 634 Cluster 0.025 10−21 110÷115 Perseus-Pisces Sup.Cluster 0.017 10−23 40÷45
OphiuchusCluster 0.028 10−21 130÷135 Abell 569 Cluster 0.019 10−23 45÷50
Columba Cluster 0.034 10−21 200÷205 Centaurus Cluster 0.020 10−23 45÷50
Hercules Sup.Cluster 0.037 10−21 205÷210 Coma Cluster 0.023 10−23 55÷60
Sculptor Sup.Cluster 0.054 10−21 340÷345 Abell 634 Cluster 0.025 10−23 60÷65
Pisces-Cetus Sup.Cluster 0.063 10−21 420÷425 Leo Sup.Cluster 0.032 10−23 85÷90
Horologium Sup.Cluster 0.067 10−21 450÷455 Hercules Sup.Cluster 0.037 10−23 100÷105
Table II: Redshifts, characteristic amplitudes, frequency Table III: Redshifts, characteristic amplitudes, frequency
range for BH-BH systems. range for WD-WDsystems.
Figure 4: Luminosity distance vs redshift for simulated BH- Figure 5: Luminosity distance vs redshift for simulated WD-
BH systems. WD systems.
asdistanceindicatorsand,possibly,asstandardcandles. signals during their inspiralling and coalescing phases.
The limits of the method are, essentially, the measure ThecoalescenceoftwoBHs,eachonewith10M⊙,could
of GW polarizations and redshifts. Besides, in order to beseenbyAdvanced-VIRGOandAdvanced-LIGOupto
improve the approach, a suitable catalogue of observed redshifts z ∼2−3.[28] Furthermore,the LISAspace in-
coalescingbinary-systemsisneeded. Thisisthemaindif- terferometer, which is expected to fly in about 10 years,
ficulty ofthe methodsince,beingthe coalescenceatran- will be sensitive to GWs in the mHz region, which cor-
sient phenomenon, it is very hard to detect and analyze responds to the waveemitted by supermassiveBHs with
the luminosity curves of these systems. Furthermore, a massesupto106M⊙. Nowadays,supermassiveBHswith
fewsimulatedsourcesareoutoftheLIGO-VIRGOband. masses between 106 and 109M⊙ are known to exist at
Next generation of interferometer (as LISA [26] or the center of most (and probably all) galaxies, including
Advanced-VIRGO and LIGO) could play a decisive role our Galaxy. The coalescence of two supermassive BHs,
to detect GWs from these systems. At the advanced which could take place, for instance during the collision
level, one expects to detect at least tens NS-NS coalesc- and merging of two galaxies or in pre-galactic structure
ing events per year,up to distances oforder2 Gpc, mea- at high redshifts, would be among the most luminous
suring the chirp mass with a precision better than 0.1%. events in the Universe. Even if the merger rate is poorly
The massesofNSs aretypicallyof order1.4M⊙. Stellar- understood, observations from the Hubble Space Tele-
mass BHs, as observed in X-ray binaries, are in general scopeandfromX-raysatellitessuchasChandra[27]have
more massive,typically with masses of order10M⊙, and revealed that these merging events could be detectable
therefore are expected to emit even more powerful GW at cosmological distances. LISA could detect them up
6
to z ∼ 10, [29, 30] and it is expected to measure sev- ment and increase the confidence of other standard can-
eral events of this kind. The most important issue that dles, [33], as well as extending the result to higher red-
can be addressed with a measure of d (z) is to under- shifts. In the future, the problem of the redshift could
L
stand “dark energy”,the quite mysterious component of be obviatefinding an electromagneticcounterpartto the
the energy budget of the Universe that manifests itself coalescence and short GRBs could play this role.
throughanaccelerationofthe expansionofthe Universe In summary, this new type of cosmic distance indi-
at high redshift. This has been observed, at z <1.7, us- cators could be considered complementary to the tra-
ing Type Ia supernovae as standard candles [31, 32]. A ditional standard candles opening the doors to a self-
possible concern in these determinations is the absence consistent gravitational astronomy.
of a solid theoretical understanding of the source. After
all, supernovae are complicated phenomena. In partic-
ular, one can be concerned about the possibility of an
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